A Modified Law of Heat Conduction of Thermoelasticity with Fractional Derivative and Relaxation Time

Abstract

(1) In the present work, a new modified thermoelasticity theory with fractional order has been constructed based on fractional calculus and Taylor series expansion of time-fractional order. The models of fractional thermoelasticity proposed by Sherief et al. [H. H. Sherief, A. M. A. El-Sayed and A. M. Abd El-Latief, Int. J. Solids Struct. 47, 269 (2010)], Ezzat [M. A. Ezzat, Phys. B 406, 30 (2011)] and Lord and Shulman with one relaxation time [H. W. Lord and Y. H. Shulman, J. Mech. Phys. Solids 15(5), 299 (1967)] as well as coupled thermoelasticity [M. A. Biot, J. Appl. Phys. 27, 240 (1956)] follow as limiting cases. This modified model is applied to an infinitely long annular cylinder. The inner and outer surfaces of the cylinder are traction free and subjected to known surrounding temperatures. Laplace transform technique will be used to get the solutions of all physical quantities. Some comparisons are shown in figures and tables to assess the effects of the fractional-order parameters in the studied fields. Results of some earlier researchers have been deduced from the current formulation. Finally, a conclusion about the new modified model has been promoted according to the analysis and numerical results.